If X is compact, then there are no ends.
- The real line R has two ends, which may be denoted ±∞. If K is a compact subset of R then by the Heine-Borel theorem K is closed and bounded. There are two unbounded components of R – K: if K is contained in the interval [a,b], they are the components containing (-∞,a) and (b,+∞). An end is a consistent choice of the left- or the right-hand component.
- The real plane R2 has one end, ∞. If K is a compact, hence closed and bounded, subset of the plane, contained in a disk, then there is a single unbounded component of R2 – K, the component containing the complement of the disc.
Denote the set of ends of X by E(X) and let . We may topologise X * by taking as neighbourhoods of an end e the sets for compact K in X.
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